Analytic semigroup

Summary

An analytic semigroup is a strongly continuous semigroup $(T(t))_{t \ge 0}$ that extends holomorphically to complex times $z$. The allowed $z$ fill a sector of the complex plane around the positive real axis. Such semigroups have a strong smoothing effect: $T(t)x$ lies in the domain of every power of the generator for $t > 0$. They are the abstract model of parabolic equations such as the heat equation. Their generators are characterised by resolvent estimates on a sector, which makes them the sectorial operators.